We have performed three-dimensional calculations of the HD$^+$
dissociation in alinearly polarized laser field with the
restriction that the nuclei are aligned with the field. Although
dissociation from a single initial vibrational state can show
substantial carrier-envelope phase (CEP) dependence, the CEP
dependence of a mixed initial state is not as strong. A strong
CEP effect can be recovered, though, by analyzing the kinetic
energy of the fragments. In order to approach realistic
experimental conditions, we perform focus volume averaging of the
results and identify the kinetic energy ranges
demonstrating the strongest CEP dependence.